To quantize GRTnon-perturbutavely we assumes that the Einstein Hilbert action from which GRTis derived is exact and not a low-energy limit of an underlying theory.Quantizing GRT without using a perturbation series yields a lot of difficulties.Three constraint equations the Hamiltonian, Gauss and diffieomorphismconstraint follow from this approach. The constraint equations form afterquantization the Wheeler-de Witt equations. These equations are highly singularand so far there are no known solutions to these equations. To circumvent thisproblem Abbhay Ashtekar introduced a new set of variables in, which today arenamed after him.
These variables turn the constraint equations into simplepolynomials. The initial hope that they would simplify the constraint equationswas damped due to the necessity of introducing a parameter in the newvariables: the Barbero-Immirzi parameter. When this parameter is chosen to becomplex, it indeed gives polynomial constraint equations.
The downside of thischoice is that it leads to a complex phase space of GRT. To obtain realsolutions reality conditions must be imposed. For the classical case this isnot a problem, but after quantizing the theory it turns out to be a majorproblem to find such reality conditions. Therefore this complex form isabandoned and the polynomial form of the constraint equations is lost.
The numerical value ofthe Barbero Immirzi parameter poses another problem.At this moment itsvalue is fixed by demanding a correct prediction of the entropy of theHawking-Berkenstein black hole. There is no physical reason for this value tobe logical.Even in the complexform problems arise when the theory is quantized. The metric is no longer asimple operator and deriving it turns out to be very complicated.
This is a firstindication that a theory, such as LQG, that uses the Ashtekar variables toquantize GRT will have difficulties finding semi-classical states. Also thequantum constraint, however simpler because of the change of the metric to thenew variables, still does not yield any results. Therefore another change ofvariables has to be made. This brings us to the loop representation.The argument for thiswas that certain functionals, loops, do annihilate the Hamiltonian constraint.They depend only on the Ashtekar variables through the trace of the holonomy, ameasure of the change of the direction of a vector when it’s paralleltransported over a closed circle (a loop).